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View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Journal of Applied Logic 6 (2008) 24–46 www.elsevier.com/locate/jal Axiomatizing geometric constructions Victor Pambuccian Department of Integrative Studies, Arizona State University, West Campus, PO Box 37100, Phoenix, AZ 85069-7100, USA Received 1 March 2005; received in revised form 14 February 2007; accepted 15 February 2007 Available online 24 February 2007 Abstract In this survey paper, we present several results linking quantifier-free axiomatizations of various Euclidean and hyperbolic geometries in languages without relation symbols to geometric constructibility theorems. Several fragments of Euclidean and hyperbolic geometries turn out to be naturally occurring only when we ask for the universal theory of the standard plane (Euclidean or hyperbolic), that can be expressed in a certain language containing only operation symbols standing for certain geometric constructions. © 2007 Elsevier B.V. All rights reserved. Keywords: Geometric constructions; Quantifier-free axiomatizations; Euclidean geometry; Absolute geometry; Hyperbolic geometry; Metric planes; Metric-Euclidean planes; Rectangular planes; Treffgeradenebenen 1. Introduction The first modern axiomatizations of geometry, by Pasch, Peano, Pieri, and Hilbert, were expressed in languages which contained, in stark contrast to the axiomatizations of arithmetic or of algebraic theories, only relation (predicate) symbols, but no operation symbol. On the other hand, geometric constructions have played an important role in geometry from the very beginning. It is quite surprising that it is only in 1968 that geometric constructions became part of the axiomatization of geometry. Two papers broke the ice: Moler and Suppes [65] and Engeler [25].In[65] we have the first axiomatization of geometry (to be precise of plane Euclidean geometry over Pythagorean ordered fields) in terms of two operations, by means of a quantifier-free axiom system. The first-order language in which it is expressed has one sort of variables, standing for points, and three individual constants a0, a1, a2, as well as two quaternary operation symbols S and I as primitive notions. The primitive notions a0, a1, a2, S and I have the following intuitive meanings: a0, a1, a2 → are three non-collinear points, S(xyuv) is a point as distant from u on the ray uv as y is from x, provided that u = v ∨ (u = v ∧ x = y), an arbitrary point, otherwise, and I(xyuv) is the point of intersection of the lines xy and uv, provided that x = y, u = v, the lines xy and uv are distinct and do intersect, an arbitrary point, otherwise. Ten years later Seeland [115] rephrased the axiom system in [65] and also gave a quantifier-free axiom system for plane Euclidean geometry over Euclidean fields, in a language enlarged with a third quaternary symbol C,having E-mail address: [email protected]. 1570-8683/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.jal.2007.02.001 V. Pambuccian / Journal of Applied Logic 6 (2008) 24–46 25 the intuitive meaning: C(xyuv) is the point of intersection of the circle centred at x and passing through y with the segment uv, provided that x = y, u lies inside and v lies outside the circle, an arbitrary point, otherwise. The difference between the Euclidean geometries over Pythagorean and Euclidean ordered fields is that the circle axiom is not assumed in the former, i.e. one does not know whether a circle and a line passing through an inner point of the circle intersect or not, whereas the latter satisfies it. Engeler’s motivation, as he states it in [26], was that, as a student of P. Bernays, he got interested in the foundations of geometry, and “rereading Hilbert’s Grundlagen der Geometrie [41] was struck by the fact that of all the topics of that book the one on geometric constructions was the least “modern”, i.e. axiomatic”.1 He thus devised a meaningful logic in which to address constructibility problems that may require a finite, but not a priori bounded, number of constructions. In this logic, we are, for example, able to determine constructively that two given segments ‘behave Archimedeanly’ (i.e. that an integer multiple of the length of either of them exceeds the length of the other), by laying off, in increasing order, integer multiples of one on the other from one of the latter’s endpoints. If we get past the endpoint of the ‘longer’ segment, we stop, if not, we continue. If the logic allows us to state that such constructions terminate after finitely many steps, then we are able to express the Archimedeanity of the coordinate field. It turns out that a quantifier-free logic, containing only Boolean combinations of halting-formulas for flow-charts (that may contain loops but not recursive calls) is all one needs. This logic was introduced by E. Engeler [21] under the name of algorithmic logic and its relevance to geometry was studied in [22–25,80,115]. It is presented in the Appendix. Such universal axiomatizations in languages without relation symbols capture the essentially constructive nature of geometry, that was the trademark of Greek geometry.2 For Proclus, who relates a view held by Geminus, “a postulate prescribes that we construct or provide some simple or easily grasped object for the exhibition of a character, while an axiom asserts some inherent attribute that is known at once to one’s auditors” [99, p. 142 (181 in the Friedlein edition)]. And “just as a problem differs from a theorem, so a postulate differs from an axiom, even though both of them are undemonstrated; the one is assumed because it is easy to construct, the other accepted because it is easy to know” [99, p. 142 (182 in the Friedlein edition)]. That is, postulates ask for the production, the πoιησ´ ις of something not yet given, of a τι, whereas axioms refer to the γνωσις˜ of a given, to insight into the validity of certain relationships that hold between given notions (cf. [30,77,137]). In traditional axiomatizations, that contain relation symbols, and where axioms are not universal statements, such as Hilbert’s, this ancient distinction no longer exists. The constructive axiomatics preserves this ancient distinction, as the ancient postulates are the primitive notions of the language, namely the individual constants and the geometric operation symbols themselves (in the Moler-Suppes case a0, a1, a2, S, I ), whereas what Geminus would refer to as “axioms” are precisely the axioms of the constructive axiom system. In the present survey, which is meant to be a guide to the relevant literature, we shall present the results obtained so far in providing quantifier-free axiomatizations in languages without relation symbols for absolute and for several Euclidean and hyperbolic two-dimensional geometries, point out the connection with classical geometric construction theorems, such as the Mohr-Mascheroni theorem (see G.E. Martin [61], L. Bieberbach [13], A. Adler [1],orGy. Szokefalvi-Nagy˝ [129] for a non-axiomatic treatment of Euclidean geometric constructions), and discuss the relevance of these axiomatizations. Languages which contain only individual constants and operation symbols as primitive notions, as well as quantifier-free axiomatizations in such languages will be called constructive. 2. Euclidean geometries Following the instruction of Voltaire’s [140] geometer, “Je vous conseille de douter de tout, excepté que les trois angles d’un triangle sont égaux à deux droits”, we will see what a progressive doubting of features of Euclidean geometry that are not related to its Euclidean metric (and thus to the sum of angles in a triangle) can lead us to. 1 A more modern treatment can be found in [49], but it is not one in which constructions become part of the language of an axiom system. A further development in the direction of more logical look at geometric constructions, somewhat similar to that in [49], can be found in [109,110]. 2 Zeuthen [146] went so far as claiming that the geometric construction was the only means of establishing the existence of a geometric object, a claim refuted in this strong form by Knorr [52] (cf. also [53]). 26 V. Pambuccian / Journal of Applied Logic 6 (2008) 24–46 A first step consists in doubting continuity, i.e. the need for R as the coordinate field of the Cartesian plane over the reals, as which standard Euclidean geometry is represented (to be referred as “the standard Euclidean plane”). This step leads to Tarski’s [114,130,132] first-order theory of the standard Euclidean plane, which turns out to be Euclidean geometry over arbitrary real-closed fields. Real-closed fields are defined as ordered fields in which (i) every positive element has a square root, and (ii) every polynomial of odd degree has a zero. Since a geometric construction instrument can be expected to provide only zeros for polynomials with degrees bounded from above, we cannot conceive of this geometry as one of Euclidean constructions with finitely many instruments. If we doubt even the Tarskian elementary form of continuity, and retain from the two conditions for real-closed fields only (i) and weaken (ii) to (ii) every polynomial of degree 3 has a zero (such fields are called in [61] Vietan fields, having been of interest to Viète) then the resulting geometry is one of geometric constructions. The instrument involved is the marked ruler (or twice-notched straightedge), which, in addition to being a ruler, allows the operation of verging or insertion or neusis (the marked ruler is a ruler with two marks (notches) on it, and verging allows, given any two intersecting lines g and h, for the positioning of the marked ruler such that one of its marks lands on g and the other lands on h).
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